The Quantum Pontiff

Proof by Logical Exhaustion

Uncertain Chad asks “What’s your favorite dubious proof technique?” I just don’t have one dubious proof technique: I have an entire book of dubious proof techniques! Seriously, I have a book where I write them all down.

But if I had to choose a dubious proof technique that was my favorite, it would have to be “proof by logical exhaustion.” Now you might think that this means that I logically list all possibilities to prove something, a technique which is perfectly valid and not very dubious at all. No, no. “Proof by logical exhaustion” is where you put forth a chain of logical reasoning so long and so tedious that your opponent is sure to get lost in the argument and in exhaustion will concede defeat.

I like this proof technique because it reminds me of a manuscript I saw once claiming to disprove special relativity which used about seven or eight changes of reference frame. I mean the argument was wrong, but showing this left you pretty near brain dead from following the reasoning (and finding the hole in the reasoning.)


  1. #1 Tyler DiPietro
    November 15, 2007

    Proof by logical exhaustion seems to be a favorite amongst cranks. Can’t overwhelm them with facts and compelling argument? Just overwhelm them with lots and lots and lots and lots….

    (5 million more “and lots”)

    …and lots of bull-poop. You’ll probably encounter it quite often from certain folks who frequent these fora.

  2. #2 Rosie Redfield
    November 15, 2007

    The little-appreciated collection of science humour called A Random Walk in Science (compiled by R. Weber in 1973) includes a fine article called The Uses of Fallacy by Paul V. Dunmore.

    It describes proof methods including reductio ad nauseam and reduction ad erratum, as well as proof by misdirection, proof by convergent irrelevancies, proof by definition, proof by assertion, proof by osmosis, and proof by aesthetics (“This result is too beautiful to be false.”).

    You can find the whole article here.

  3. #3 D. Eppstein
    November 16, 2007

    So does this also cover the one where you start doing a long tedious case analysis, get tired of it partway through, and end with a statement that the rest of the cases are similar?

  4. #4 Koray
    November 16, 2007

    Or you can prove a rather long but mechanical lemma in full detail, tiring the reader, and then claim a more complicated fallacy with “Proof: Obvious.” and follow that with another lengthy proof. Bonus point: it may make some people feel like they’re idiots for not seeing what’s so bloody obvious.

    The one I like is the one where you pull out references to all conceivable obscure fields, trying to gauge which field your opponent is least familiar with from his responses, and then divert the discussion into that field. Unless he’s come with an encyclopedia under his arm, that discussion is at least a tie.

  5. #5 Blake Stacey
    November 16, 2007

    I’ve always liked “proof by changing all the 2s to ns”, “proof by rotating both sides of the equation”, and “integration by court order”.

  6. #6 guthrie
    November 16, 2007

    Rosie, I don’t see a working link to the article, which sounds interesting.

    This sort of thing is why I am very suspicious of philosophers and others of similar bent. Me, I say “reason how you will, but I want to be able to bang things together to see if they can survive”.

  7. #7 PK
    November 16, 2007

    My favourite is “proof by intimidation” when I show a slide full of equations (or difficult graphs). I think I first heard this phrase from Jon Dowling.

  8. #8 ScentOfViolets
    November 16, 2007

    I think this is it

  9. #9 Alastair McKinstry
    November 17, 2007

    Proof by exhaustion reminds me of a comment by a lecturer of mine in college, Tim Murphy, said of “Nicolas Bourbaki”s proofs:

    If, after 200 pages of proof, you’ve shown that 2 + 2 = 5,
    what do you believe, that (a) its 5, or (b), you’ve made a
    mistake somewhere? And if (b), why did you bother doing
    it in the first place?

  10. #10 terrence
    November 19, 2007

    I think I sat through several “proof by logical exhaustion” in various college classes. I generally preferred it when my professors stuck with “proof by one example.”

  11. #11 PK
    November 19, 2007

    how about redctio ad hitlerum?

  12. #12 andy.s
    November 19, 2007

    Fermat-style proof?
    i.e., I have a proof that this margin is too small to contain.

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